Last week Heron’s formula came up in the post An Unexpected Triangle. Given the lengths of the sides of a triangle, there is a simple expression for the area of the triangle.
where the sides are a, b, and c and s is the semiperimeter, half the perimeter.
Is there an analogous formula for the area of a quadrilateral? Yes and no. If the quadrilateral is cyclic, meaning there exists a circle going through all four of its vertices, then Brahmagupta’s formula for the area of a quadrilateral is a direct generalization of Heron’s formula for the area of a triangle. If the sides of the cyclic quadrilateral are a, b, c, and d, then the area of the quadrilateral is
where again s is the semiperimeter.
But in general, the area of a quadrilateral is not determined by the length of its sides alone. There is a more general expression, Bretschneider’s formula, that expresses the area of a general quadrilateral in terms of the lengths of its sides and the sum of two opposite angles. (Either pair of opposite angles add to the same value.)
In a cyclic quadrilateral, the opposite angles α and γ add up to π, and so the cosine term drops out.
As written, you’ve got an extra s-factor under the radical Brahmagupta’s formula (You can tell somethings wrong b/c it doesn’t make sense dimensionally).